On the Existence of Eigenvalues of a Three Particle System on a Cubic Lattice

We consider a three particle system on the lattice Zᵈ , $$d=1,2$$ , consisting of two identical bosons and one other particle interacting via pairwise attractive contact potentials. For a fixed K∈ Tᵈ , the system is described by the fiber Schrödinger operator Hλμ(K) , where λ<0 and μ<0 , respectively, characterize the interaction of two bosons and the interaction of a boson with another particle, and K denotes the total quasi-momentum of the system. The existence of an eigenvalue of the operator Hλμ(K) below the lower boundary of its essential spectrum is proved for any λ,μ<0 . Furthermore, the analytical dependence of the eigenvalue and the corresponding eigenfunction on the parameter K∈ Tᵈ is established.